Verify that every member of the family of functions y = (ln(x) + C) / (x) is a s

Question

Verify that every member of the family of functions y = (ln(x) + C) / (x) is a solution to the differential equation x^2*y' + xy = 1.

1. Find a solution that satisfies the initial condition y(9)=2.

2. Find a solution that satisfies the initial condition y(2)=9.


Thanks for any help in getting me going on this problem. In verifying that the solution y=(ln(x)+C)/(x) I plugged it in to the differential equation and worked it down to ln(x) + C =1. So indeed I think that every x greater than 0 will work. I just am not sure what I need to do for 1 and 2, it seems to me like the way I should do it is incorrect.

Explanation / Answer

Solve the linear equation x^2 ( dy(x))/( dx)+x y(x) = 1: Divide both sides by x^2: ( dy(x))/( dx)+(y(x))/x = 1/x^2 Let mu(x) = exp( integral 1/x dx) = x. Multiply both sides by mu(x): x ( dy(x))/( dx)+y(x) = 1/x Substitute 1 = ( d)/( dx)(x): x ( dy(x))/( dx)+( d)/( dx)(x) y(x) = 1/x Apply the reverse product rule f ( dg)/( dx)+( df)/( dx) g = ( d)/( dx)(f g) to the left-hand side: ( d)/( dx)(x y(x)) = 1/x Integrate both sides with respect to x: integral ( d)/( dx)(x y(x)) dx = integral 1/x dx Evaluate the integrals: x y(x) = log(x)+c_1, where c_1 is an arbitrary constant. Divide both sides by mu(x) = x: y(x) = (log(x)+c_1)/x 1.)y(x) = (log(x/9)+18)/x 2. y(x) = (log(x/2)+18)/x

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